Editing Adele ring (section)

==Examples==

=== Ring of adeles for the rational numbers ===
The rationals <math>K = \bold{Q}</math> have a valuation for every prime number <math>p</math>, with <math>( K_\nu,\mathcal{O}_\nu)=(\mathbf{Q}_p,\mathbf{Z}_p)</math>, and one infinite valuation ''∞'' with <math>\mathbf{Q}_\infty = \mathbf{R}</math>. Thus an element of
:<math>\mathbf{A}_\mathbf{Q}\ = \ \mathbf{R}\times \prod_p (\mathbf{Q}_p,\mathbf{Z}_p)</math>
is a real number along with a [[P-adic number|''p''-adic]] rational for each ''<math>p</math>'' of which all but finitely many are ''p''-adic integers.

=== Ring of adeles for the function field of the projective line ===
Secondly, take the function field <math>K=\mathbf{F}_q(\mathbf{P}^1)=\mathbf{F}_q(t)</math> of the [[projective line]] over a finite field. Its valuations correspond to points <math>x</math> of <math>X=\mathbf{P}^1</math>, i.e. maps over <math>\text{Spec}\mathbf{F}_{q}</math>
:<math>x\ :\ \text{Spec}\mathbf{F}_{q^n}\ \longrightarrow \ \mathbf{P}^1.</math>
For instance, there are <math>q+1</math> points of the form <math>\text{Spec}\mathbf{F}_{q}\ \longrightarrow \ \mathbf{P}^1</math>. In this case <math>\mathcal{O}_\nu=\widehat{\mathcal{O}}_{X,x}</math> is the [[Completion of a ring|completed]] stalk of the [[structure sheaf]] at <math>x</math> (i.e. functions on a formal neighbourhood of <math>x</math>) and <math>K_\nu=K_{X,x}</math> is its fraction field. Thus
:<math>\mathbf{A}_{\mathbf{F}_q(\mathbf{P}^1)}\ =\ \prod_{x\in X} (\mathcal{K}_{X,x},\widehat{\mathcal{O}}_{X,x}).</math>
The same holds for any smooth proper curve <math>X/\mathbf{F_{\mathit{q}}}</math> over a finite field, the restricted product being over all points of <math>x \in X</math>.